×

zbMATH — the first resource for mathematics

A waiting time phenomenon for thin film equations. (English) Zbl 1024.35051
The authors consider the fourth-order degenerate parabolic equation \[ u_t + \nabla \cdot (|u|^n \nabla \Delta u)=0, \quad t>0, \;x \in \Omega, \tag{1} \] where \(\Omega \subseteq \mathbb{R}^N\), \(N \in \{ 1,2,3\}\). Equation (1) is supplemented with suitable initial condition \(u_0\) with compact support strictly contained in \(\Omega\) and Neumann boundary conditions on \(u\) and \(\Delta u\) on \(\partial \Omega\). Specifically, for \(n \in (0,3)\) they ask the following question: Can one formulate conditions on the initial data \(u_0\) so that the solution of (1), \(u(t)\) exhibits the waiting time phenomenon, which in the case of \(N=1\) means the following: Let \(x_0 \in \partial \operatorname {supp} u_0\). Then \(u(t)\) exhibits the waiting time phenomenon at \(x_0\) if there is a time \(T^*\) and a neighbourhood \(B(x_0)\) of \(x_0\) such that \(\operatorname {supp} u(t) \cap B(x_0) = \operatorname {supp} u_0 \cap B(x_0)\); for the multidimensional version (employing an external cone condition) see section 5 of the paper.
The main results of the paper are: For \(0<n<2\) and \(N=1\) or if \(1/8<n<2\), \(N=2,3\), the waiting time phenomenon occurs at a point \(x_0 \in \partial\operatorname {supp} u_0\) if \(u_0(x)\) grows at most as \(|x-x_0|^{4/n}\) in a neighbourhood of \(x_0\) (Theorems 4.1 and 5.1). For \(2\leq n < 3\), \(N=1\) the waiting time phenomenon occurs at \(x_0\) if \(u_0x\) grows ar most like \(|x-x_0|^{4/n-1}\) (Theorem 6.1). Finally, in section 7 optimality of these exponents is considered and results of numerics suggesting such optimality are presented.
The main tools used in the proofs are a version of the Gagliardo-Nirenberg inequalities (Theorem 2.3), entropy (for \(n<2\)) and weighted energy estimates (for \(n \geq 2\)), and a crucial extension of an iteration lemma due to Stampacchia (Lemma 3.1).

MSC:
35K65 Degenerate parabolic equations
35K35 Initial-boundary value problems for higher-order parabolic equations
35R35 Free boundary problems for PDEs
76A20 Thin fluid films
74K35 Thin films
PDF BibTeX XML Cite
Full Text: Numdam EuDML
References:
[1] N.D. Alikakos , On the pointwise behavior of the solutions of the porous medium equation as t approaches zero or infinity , Nonlinear Anal. 9 ( 1985 ), 1095 - 1113 . MR 806912 | Zbl 0589.35064 · Zbl 0589.35064
[2] D.G. Aronson , ” The porous medium equation ”, In A. Dold and B. Eckmann, editors, Nonlinear Diffusion Problems . Lecture Notes in Mathematics , 1224 , Springer-Verlag , 1985 . MR 877986 | Zbl 0626.76097 · Zbl 0626.76097
[3] E. Beretta - M. Bertsch - R. Dal Passo , Nonnegative solutions of a fourth order nonlinear degenerate parabolic equation , Arch. Rat. Mech. Anal. 129 ( 1995 ), 175 - 200 . MR 1328475 | Zbl 0827.35065 · Zbl 0827.35065
[4] F. Bernis , Viscous flows, fourth order nonlinear degenerate parabolic equations and singular elliptic problems , In:” Free boundary problems: theory and applications ”, J. I. Diaz - M. A. Herrero - A. Linan - J. L. Vazquez (eds.), Pitman Research Notes in Mathematics 323 , Longman , Harlow , 1995 , pp. 40 - 56 . MR 1342325 | Zbl 0839.35102 · Zbl 0839.35102
[5] F. Bernis , Finite speed ofpropagation and continuity of the interfacefor thin viscous flows , Adv. Differential Equations 1 no. 3 ( 1996 ), 337 - 368 . MR 1401398 | Zbl 0846.35058 · Zbl 0846.35058
[6] F. Bernis , Finite speed of propagation for thin viscous flows when 2 \leq n &lt; 3 , C.R. Acad. Sci. Paris Sér. I Math. 322 ( 1996 ). Zbl 0853.76018 · Zbl 0853.76018
[7] F. Bernis - A. Friedman , Higher order nonlinear degenerate parabolic equations , J. Differential Equations 83 ( 1990 ), 179 - 206 . MR 1031383 | Zbl 0702.35143 · Zbl 0702.35143
[8] F. Bernis - L.A. Peletier - S.M. Williams , Source-type solutions of a fourth order nonlinear degenerate parabolic equations , Nonlinear Anal. 18 ( 1992 ), 217 - 234 . MR 1148286 | Zbl 0778.35056 · Zbl 0778.35056
[9] A. Bertozzi - M. Pugh , The lubrication approximation for thin viscous films: the moving contact line with a porous media cut off of van der waals interactions , Nonlinearity 7 ( 1994 ), 1535 - 1564 . MR 1304438 | Zbl 0811.35045 · Zbl 0811.35045
[10] A.L. Bertozzi - M. Pugh , The lubrication approximation for thin viscous films: regularity and long time behaviour of weak solutions , Nonlinear Anal. 18 ( 1992 ), 217 - 234 .
[11] M. Bertsch - R. Dal Passo - H. Garcke - G. Grün , The thin viscous flow equation in higher space dimensions , Adv. Differential Equations 3 ( 1998 ), 417 - 440 . MR 1751951 | Zbl 0954.35035 · Zbl 0954.35035
[12] R. Dal Passo - H. Garcke , Solutions of a fourth order degenerate parabolic equation with weak initial trace , Ann. Scuola Norm. Sup. Pisa Cl. Sci . ( 4 ) 28 ( 1999 ), 153 - 181 . Numdam | MR 1679081 | Zbl 0945.35049 · Zbl 0945.35049
[13] R. Dal Passo - H. Garcke - G. Grün , On a fourth order degenerate parabolic equation: global entropy estimates and qualitative behaviour of solutions , SIAM J. Math. Anal. 29 ( 1998 ), 321 - 342 . MR 1616558 | Zbl 0929.35061 · Zbl 0929.35061
[14] R. Dal Passo - L. Giacomelli - A. Shishkov , The thin film equation with nonlinear diffusion , Preprint Me.Mo.Mat. Department 2/2000, to appear in Comm. Partial Differential Equations . MR 1865938 | Zbl 1001.35070 · Zbl 1001.35070
[15] E.B. Dussan - S. Davis , On the motion of a fluid-fluid interface along a solid surface , J. Fluid Mech. 65 ( 1974 ), 71 - 95 . Zbl 0282.76004 · Zbl 0282.76004
[16] R. Ferreira - F. Bernis , Source-type solutions to thin-film equations in higher space dimensions , European J. Appl. Math. 8 ( 1997 ), 507 - 524 . MR 1479525 | Zbl 0894.76019 · Zbl 0894.76019
[17] E. Gagliardo , Ulteriori properità di alcune classi di funzioni in piú variabili , Ricerche di Mat . ( 1959 ), 24 - 51 . MR 109295 | Zbl 0199.44701 · Zbl 0199.44701
[18] G. Grün , Degenerate parabolic equations of fourth order and a plasticity model with nonlocal hardening , Z. Anal. Anwendungen 14 ( 1995 ), 541 - 573 . MR 1362530 | Zbl 0835.35061 · Zbl 0835.35061
[19] G. Grün - M. Rumpf , Nonnegativity preserving convergent schemes for the thin film equation , Numer. Mathematik 87 ( 2000 ), 113 - 152 . MR 1800156 | Zbl 0988.76056 · Zbl 0988.76056
[20] G. Grün - M. Rumpf , Entropy consistent finite volume schemes for the thin film equation , In: ” Finite volume schemes for complex applications II ”, R. Vilsmeier - F. Benkhaldoun - D. Hänel (eds.), Hermes Science Publications , Paris , 1999 , pp. 205 - 214 . MR 2062140 | Zbl 1052.65526 · Zbl 1052.65526
[21] J. Hulshof - A. Shishkov , The thin film equation with 2 \leq n &lt; 3: finite speed of propagation in terms of the l1-norm , Adv. Differential Equations 3 ( 1998 ), 625 - 642 . Zbl 0953.35072 · Zbl 0953.35072
[22] B.F. Knerr , The porous medium equation in one dimension , Trans. Amer. Math. Soc. 234 ( 1977 ), 381 - 415 . MR 492856 | Zbl 0365.35030 · Zbl 0365.35030
[23] L. Nirenberg , On elliptic partial differential equations , Ann. Scuola Norm. Sup. Pisa, Cl. Sci. 13 ( 1959 ), 115 - 162 . Numdam | MR 109940 | Zbl 0088.07601 · Zbl 0088.07601
[24] A. Oron - S.H. Davis - S.G. Bankoff , Long-scale evolution of thin liquid films , Reviews of Modem Physics 69 ( 1997 ), 932 - 977 .
[25] F. Otto , Lubrication approximation with prescribed non-zero contact angle: an existence result , Comm. Partial Differential Equations 23 ( 1998 ), 2077 - 2164 . MR 1662172 | Zbl 0923.35211 · Zbl 0923.35211
[26] N.F. Smyth - J.M. Hill , Higher order nonlinear diffusion , IMA J. Applied Mathematics 40 ( 1988 ), 73 - 86 . MR 983990 | Zbl 0694.35091 · Zbl 0694.35091
[27] G. Stampacchia , ”Équations elliptiques du second ordre à coefficients discontinus” , Les presses de l’université de Montréal , Montréal , 1966 . MR 251373 | Zbl 0151.15501 · Zbl 0151.15501
[28] J.L. Vazquez , An introduction to the mathematical theory of the porous medium equation, In: ”Shape Optimization and Free Boundaries” , M. C. Delfour-G. Sabidussi (eds.), Kluwer Academic Publishers , Netherlands, 1992 , pp. 347 - 389 . MR 1260981 | Zbl 0765.76086 · Zbl 0765.76086
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.